nLab
Gelfand spectrum

Contents

Idea

The Gelfand spectrum (originally Гельфанд) of a commutative C*-algebra AA is a topological space XX such that AA is the algebra of complex-valued continuous functions on XX. (This “Gelfand duality” is a special case of the general duality between spaces and their algebras of functions.)

Definition

Definition

Given a unital, not necessarily commutative, complex C*-algebra AA, the set of its characters, that is: continuous nonzero linear homomorphisms into the field of complex numbers, is canonically equipped with what is called the spectral topology which is compact Hausdorff. (If applied to a nonunital C *C^*-algebra, then it is only locally compact.) This correspondence extends to a functor, called the Gel’fand spectrum from the category C*Alg of unital C *C^*-algebras to the category of Hausdorff topological spaces.

Remark

A character on a unital Banach algebra is automatically a continuous function (with Lipschitz constant 1).

Remark

The Gelfand spectrum functor is a full and faithful functor when restricted to the subcategory of commutative unital C *C^*-algebras.

Remark

The kernel of a character is clearly a codimension-11 closed subspace, and in particular a closed maximal ideal in AA; therefore the Gel’fand spectrum is a topologised analogue of the maximal spectrum of a discrete algebra.

Remark

For noncommutative C *C^*-algebras the spaces of equivalence classes of irreducible representations (i.e., the spectrum) and their kernels (i.e., the primitive ideal space) are more important than the character space.

Remark

The Gelfand spectrum is also useful in the context of more general commutative Banach algebras.

category: analysis

Revised on November 21, 2013 11:37:59 by Urs Schreiber (188.200.54.65)